| L(s) = 1 | − 2-s + 3-s − 6-s + 8-s + 3·9-s + 11-s − 10·13-s − 16-s + 6·17-s − 3·18-s + 2·19-s − 22-s − 6·23-s + 24-s + 5·25-s + 10·26-s + 8·27-s + 6·29-s + 8·31-s + 33-s − 6·34-s − 2·37-s − 2·38-s − 10·39-s + 12·41-s − 8·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 0.408·6-s + 0.353·8-s + 9-s + 0.301·11-s − 2.77·13-s − 1/4·16-s + 1.45·17-s − 0.707·18-s + 0.458·19-s − 0.213·22-s − 1.25·23-s + 0.204·24-s + 25-s + 1.96·26-s + 1.53·27-s + 1.11·29-s + 1.43·31-s + 0.174·33-s − 1.02·34-s − 0.328·37-s − 0.324·38-s − 1.60·39-s + 1.87·41-s − 1.21·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.736390332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.736390332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12 T + 91 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 13 T + 102 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.07701642052156932906872840343, −9.891313535945065295029212240653, −9.276068965008925856115601144395, −8.802081053915098413955511296077, −8.608413977034276326856076357879, −7.908602569556590154192815388525, −7.55877263803316820089936227582, −7.44031222685463315740148897000, −6.98801305387539476044862840532, −6.43149051878647590513864781921, −5.93762077663309293568934369421, −5.09332606534080148262337780558, −4.99221281539543737110989762841, −4.32283077678062095143029995430, −4.09801734957459746943317776448, −3.00373579663891892745306727336, −2.85166500700558759535149286656, −2.22657022390552177743775951526, −1.36908007570138414945336054446, −0.69902140681725652244865042197,
0.69902140681725652244865042197, 1.36908007570138414945336054446, 2.22657022390552177743775951526, 2.85166500700558759535149286656, 3.00373579663891892745306727336, 4.09801734957459746943317776448, 4.32283077678062095143029995430, 4.99221281539543737110989762841, 5.09332606534080148262337780558, 5.93762077663309293568934369421, 6.43149051878647590513864781921, 6.98801305387539476044862840532, 7.44031222685463315740148897000, 7.55877263803316820089936227582, 7.908602569556590154192815388525, 8.608413977034276326856076357879, 8.802081053915098413955511296077, 9.276068965008925856115601144395, 9.891313535945065295029212240653, 10.07701642052156932906872840343
